q1 — pure Example 4 on the {b,1} ray

The 2026-08-17 campaign certified 0.38285 at mix weight β=1/5 and stopped the β scan at 0.40. On this ray the first-crossing keeps rising through β=1.

For β=1 and b in (1−1/√2, 1/2], Liu Example 4 saturates Π_{b,b}(0,0)=1/2, so the equality mean is 1−(1−b)h(b). The minimum solves

h(b) = (1−b) log2((1−b)/b)

and equals 0.3830513565868…. The claimed 5-decimal constant is 0.38304, strictly below that crossing and strictly above 0.38285.

Replay:

./run_all.sh

That is: solve_crossing.py (mpmath dps=80), verify.py (published constants, analytic residual, 4500×3500 mesh), and verify.c (same mesh, nested loop). Certificate: certs/verify.json.

This is the Gilmer constant for the optimizer family {b,1}, the same hypothesis class as Liu Theorem 13. It is not 1/2, and it is not a bound for every measure on [0,1]. Isolated 2-atomic samples live in certs/hunt_two_atomic.json and are not a lower bound.